en>Shabble: /* Examples */ Adding link to Reed–Solomon error correction

2014-11-21T13:23:32Z

Examples: Adding link to Reed–Solomon error correction

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	~~In [[probability theory]], '''Etemadi~~'s ~~inequality''' is a so-called "maximal inequality", an [[inequality (mathematics)\|inequality]] that gives a bound on~~ the ~~[[probability]] that the [[partial sum]]s of a [[Finite set\|finite]] collection of [[independent random variables]] exceed some specified bound~~. ~~The result~~ is ~~due~~ to ~~[[Nasrollah Etemadi]]~~.		The name of the author is Figures but it's not the most masucline title out there. Hiring is my occupation. His family life in South Dakota but his spouse wants them to transfer. Doing ceramics is what love performing.<br><br>my web-site at home std test ([http://www.buzzbit.net/user/AFilson click through the up coming website page])

	~~==Statement of the inequality==~~

	~~Let ''X''<sub>1</sub>,~~ .~~.., ''X''~~<~~sub~~>~~''n''~~<~~/sub~~> ~~be independent real~~-~~valued random variables defined on some common~~ [~~[probability space]], and let ''α'' ≥ 0. Let ''S''<sub>''k''<~~/~~sub> denote the partial sum~~

	~~:<math>S_{k} = X_{1} + \cdots + X_{k}~~.~~\,</math>~~

	~~Then~~

	~~:<math>\mathbb{P} \left( \max_{1 \leq k \leq n} \| S_{k} \| \geq 3 \alpha \right) \leq 3 \max_{1 \leq k \leq n} \mathbb{P} \left( \| S_{k} \| \geq \alpha \right)~~.</~~math>~~

	~~==Remark==~~

	~~Suppose that the random variables ''X''<sub>''k''<~~/~~sub> have common [[expected value]] zero. Apply [[Chebyshev's inequality]] to~~ the ~~right-hand side of Etemadi's inequality and replace ''α'' by ''α'' / 3. The result is [[Kolmogorov's inequality~~]~~] with an extra factor of 27 on the right-hand side:~~

	~~:<math>\mathbb{P} \left( \max_{1 \leq k \leq n} \| S_{k} \| \geq \alpha \right) \leq \frac{27}{\alpha^{2}} \mathrm{Var} (S_{n}~~)~~.</math>~~

	~~==References==~~

	* {{cite book \| last=Billingsley \| first=Patrick \| title=Probability and Measure \| publisher=John Wiley & Sons, Inc. \| location=New York \| year=1995 \| isbn=0-471-00710-2}} (Theorem 22.5)
	* {{cite journal \| last=Etemadi \| first=Nasrollah \| title=On some classical results in probability theory \| journal=[[Sankhya (journal)\|Sankhyā]] Ser. A \| volume=47 \| year=1985 \| pages=215–221 \|mr=0844022 \| jstor = 25050536 \| issue=2 }}

	~~[[Category:Probabilistic inequalities]]~~
	~~[[Category:Statistical inequalities]]~~

en>EleGu79: /* Properties */

2013-05-14T20:00:16Z

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en>Xqbot: r2.7.3) (Robot: Removing he:קוד סיסטמטי

2012-06-28T06:17:01Z

r2.7.3) (Robot: Removing he:קוד סיסטמטי

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en>Shabble: /* Examples */ Adding link to Reed–Solomon error correction

en>EleGu79: /* Properties */

en>Xqbot: r2.7.3) (Robot: Removing he:קוד סיסטמטי